Subexponential Parameterized Algorithms for Hitting Subgraphs

TL;DR

This work develops a general framework for subexponential parameterized algorithms for the finite-structure hitting problem $\mathcal{F}$-Hitting on broad graph classes that admit balanced separators with $\rho<1$. The core technical advance is a subexponential branching scheme built around heavy cores and sunflower-based witnesses, combined with a linear-time kernel and a reduction to hitting-set instances whose Gaifman graphs have sublinear treewidth. Each instance can then be solved by treewidth-based dynamic programming, yielding a running time of $2^{O(k^c)}\cdot n + O(m)$ with $c<1$. The framework applies to weighted versions and to graph families such as polynomial-expansion and many geometric intersection graphs, significantly broadening the scope of subexponential FPT results. The results impact a wide range of vertex-deletion problems under a unified, scalable approach with potential for further extensions to induced variants and tighter bounds.

Abstract

For a finite set $\mathcal{F}$ of graphs, the $\mathcal{F}$-Hitting problem aims to compute, for a given graph $G$ (taken from some graph class $\mathcal{G}$) of $n$ vertices (and $m$ edges) and a parameter $k\in\mathbb{N}$, a set $S$ of vertices in $G$ such that $|S|\leq k$ and $G-S$ does not contain any subgraph isomorphic to a graph in $\mathcal{F}$. As a generic problem, $\mathcal{F}$-Hitting subsumes many fundamental vertex-deletion problems that are well-studied in the literature. The $\mathcal{F}$-Hitting problem admits a simple branching algorithm with running time $2^{O(k)}\cdot n^{O(1)}$, while it cannot be solved in $2^{o(k)}\cdot n^{O(1)}$ time on general graphs assuming the ETH. In this paper, we establish a general framework to design subexponential parameterized algorithms for the $\mathcal{F}$-Hitting problem on a broad family of graph classes. Specifically, our framework yields algorithms that solve $\mathcal{F}$-Hitting with running time $2^{O(k^c)}\cdot n+O(m)$ for a constant $c<1$ on any graph class $\mathcal{G}$ that admits balanced separators whose size is (strongly) sublinear in the number of vertices and polynomial in the size of a maximum clique. Examples include all graph classes of polynomial expansion and many important classes of geometric intersection graphs. Our algorithms also apply to the \textit{weighted} version of $\mathcal{F}$-Hitting, where each vertex of $G$ has a weight and the goal is to compute the set $S$ with a minimum weight that satisfies the desired conditions. The core of our framework is an intricate subexponential branching algorithm that reduces an instance of $\mathcal{F}$-Hitting (on the aforementioned graph classes) to $2^{O(k^c)}$ general hitting-set instances, where the Gaifman graph of each instance has treewidth $O(k^c)$, for some constant $c<1$ depending on $\mathcal{F}$ and the graph class.

Theorems & Definitions (50)

• Theorem 1.1
• Theorem 1.1
• Theorem 1.1
• Corollary 1.2
• proof
• Lemma 2.1: simplified version of Lemma \ref{['lem-twGaifman']}
• Lemma 2.2: simplified version of Lemma \ref{['lem-heavysep']}
• proof : Proof sketch.
• proof : Proof sketch.
• Lemma 3.1